2009 AMC 12B Problem 24

For how many values of x in [0,\pi] is \sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)? Note: The functions \sin^{ - 1} = \arcsin and \cos^{ - 1} = \arccos denote inverse trigonometric functions.

\textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.